Towards the exact nature of a certain error term. II (Q1202432)

Let \(k\) be a squarefree natural number and define \(\delta_ k(n)=\max(d: d| n,\;(d,k)=1\}\). If \(E_ k(x)\) is the error term defined by the following formula \[ \sum_{n\leq x}\delta_ k(n)={k x^ 2\over 2\sigma(k)}+E_ k(x),\;\text{ where }\sigma(k)=\sum_{d| k}d, \] to find nice bounds for \(\limsup_{x\to\infty}E_ k(x)/x\) and \(\liminf_{x\to\infty}E_ k(x)/x\) does not seem to be easy. The problem had been taken up by several authors; for example, \textit{J. Herzog} and \textit{Th. Maxsein} [Arch. Math. 50, No. 2, 145-155 (1988; Zbl 0616.10035)], \textit{S. D. Adhikari} and \textit{R. Balasubramanian} [Arch. Math. 56, 37-40 (1988; Zbl 0686.10033)], \textit{Y.-F. S. Pétermann} [Enseign. Math., II. Sér. 37, No. 3/4, 213-222 (1991)]. In the present paper, among other results it is proved that if \(k=p_ 1p_ 2\dots p_ r\), where all the primes \(p_ i\) are of the form \(-1\) \(\pmod s\) for some \(s>2\), then, \[ \limsup_{x\to \infty}E_ k(x)/x\geq 2^{r-1}(1-2/s)\prod_{p_ i| k}\bigl(1-{1\over 2(p_ i+1)}\bigr), \] \[ \liminf_{x\to\infty}E_ k(x)/x\leq -2^{r-1}(1- 2/s)\prod_{p_ i| k}\bigl( 1-{1\over 2(p_ i+1)}\bigr). \] The authors also make the following conjecture: \[ \limsup_{x\to \infty}{| E_ k(x)|\over x}\leq 2^{r-1}\prod_{p_ i| k}\bigl( 1-{1\over 2(p_ i+1)}\bigr). \]